Icosahedral Packing of RNA Viral Genomes

Many spherelike RNA viruses package a portion of their genome in a manner that mirrors the icosahedral symmetry of the protein container, or capsid. Graph-theoretical constraints forbid exact realization of icosahedral symmetry. This paper explores the consequences of graph-theoretical constraints on quasi-icosahedral genome structures. A key result is the prediction that the genome organization is a Hamiltonian path or cycle and that the associated assembly scenario of such single-stranded spherelike RNA viruses resembles that of cylindrical RNA viruses, such as tobacco mosaic viruses.

Figure 1

Image of the pariacoto virus, reconstructed from cryoelectron microscopy by Tang et al. [11]. The RNA is shown as a ribbon.

Figure 2

Two possible secondary structures for the Pariacoto RNA2 molecule. Structure (A) possesses the lowest free energy. Structure (B) is a (nearly) linearly branched structure only $5k_{B}T$ higher in energy than that of structure (A). The backbones are indicated by dashed curves.

Figure 3

Vertex types. (A1) ("right turn") and (A2) ("left turn") are pure branch points related by a rotation over 180 degrees along one of their branches. The arrows indicate the $3^{\prime }$ to $5^{\prime }$ direction of the RNA molecule. Note that electrostatic repulsion favors $\sim 120°$ angles between the branches. Type (B) vertex consists of a combination bubble/hairpin, with a total of six variations. The presence of the bubble permits a sharp kink between the two connected duplex sections required at a vertex of the dodecahedron. The electrostatic energy cost of a (B) vertex is expected to exceed that of an (A) vertex. Type (C) vertex: three hairpin/stem loops are located at one vertex. A (C) vertex requires that six RNA strands are located at a single vertex, and hence should have the highest energy cost.

Figure 4

Two examples of modified Euler paths covering the edges of a dodecahedron. The vertex at which each path begins and ends is in the upper central portion of the dodecahedral graph. The path at the left contains the minimum number of crossings, while the path at the right contains the maximum number of crossings.

Figure 5

Assembly of a viral capsid governed by coordinating it with the tracing out of a Hamiltonian path by the icosahedrally ordered RNA. For the fully assembled capsid, shown in the lower right-hand corner of the figure, all portions of the Hamiltonian path traced out by the RNA main chain are indicated.